Properties

Label 2.531.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $531$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(531\)\(\medspace = 3^{2} \cdot 59 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.93987.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.59.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 9 a + 11 + a\cdot 13 + 3\cdot 13^{2} + \left(4 a + 11\right)\cdot 13^{3} + \left(9 a + 6\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 a + 11 + a\cdot 13 + \left(8 a + 10\right)\cdot 13^{2} + \left(4 a + 2\right)\cdot 13^{3} + \left(4 a + 6\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 5\cdot 13 + 8\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 7 + \left(11 a + 6\right)\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + \left(8 a + 2\right)\cdot 13^{3} + \left(3 a + 12\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 8 + \left(11 a + 5\right)\cdot 13 + \left(4 a + 3\right)\cdot 13^{2} + \left(8 a + 12\right)\cdot 13^{3} + \left(8 a + 5\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 + 6\cdot 13 + 12\cdot 13^{2} + 10\cdot 13^{3} +O(13^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(2,6)(3,4)$
$(1,2)(3,6)(4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,5)(2,4)(3,6)$ $-2$
$3$ $2$ $(1,2)(3,6)(4,5)$ $0$
$3$ $2$ $(1,3)(5,6)$ $0$
$2$ $3$ $(1,4,3)(2,6,5)$ $-1$
$2$ $6$ $(1,6,4,5,3,2)$ $1$
The blue line marks the conjugacy class containing complex conjugation.